Yes. Private schools do allow students to enroll as 9th graders even if they have completed 9th grade.
You are not holding this child back. You are giving them the gift of time.
I am a big fan of giving the gift of time. My DH repeated 11th grade. And yea, this was at a private international school abroad. It was a great decision, he says. He completed 11th grade in a previous country. When he enrolled, the headmaster politely said he could be in 12th grade, but it would be better if he did 2 years at their school. To this day…DH says things clicked a lot better.
We had a November birthday kid, and our state at the time allowed enrollment if a child was 5 by January 1. We elected to have the kid do a PG year in preschool. It worked out very well. And many years later, the kid remarked that it was the right decision.